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- W2039219076 abstract "A formula for the derivative with respect to a parameter $nu $ of a zero of a suitable solution $z(t,nu )$ of the differential equation $(p(t,nu )z')' + r(t,nu )z = 0$ is derived. This provides a kind of quantitative confirmation of the Sturm comparison theorem in that it shows that the zeros are monotonic in $nu $ in the same direction as $p(t,nu )$ and in a direction opposite to that of $r(t,nu )$. It is also shown that the derivative of a zero with respect to a parameter is equal to the value at the zero of a function that satisfies a third-order linear nonhomogeneous differential equation. Different methods of solving this equation lead to different formulas for the derivative. The results are applied to get formulas for the derivative with respect to $lambda $ of the zeros of the Gegenbauer polynomial $P_n^{(lambda )} (x)$. The method also yields a generalization to arbitrary cylinder functions of formulas due, respectively, to Schläfli and Schafheitlin for the derivative with respect to order of the zeros of the Bessel functions $J_nu (x)$ and $Y_nu (x)$." @default.
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- W2039219076 date "1994-03-01" @default.
- W2039219076 modified "2023-10-17" @default.
- W2039219076 title "On the Derivative with Respect to a Parameter of a Zero of a Sturm–Liouville Function" @default.
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- W2039219076 doi "https://doi.org/10.1137/s0036141092228878" @default.
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